When a magnetic field is applied perpendicular to a current flowing in a finite size semiconducting material, the combination of the current and magnetic field produces a Lorentz force on carriers within the semiconducting material. This force pushes the carriers into circular paths around the magnetic field lines. The carriers are constrained within the material, which creates an electric field perpendicular to both the current and the magnetic field. The voltage produced by this electric field is called the Hall effect voltage, or Hall voltage.
A steady state condition is reached when the force from the transverse electric field exactly cancels the Lorentz force. Accordingly, in this state the Hall voltage is proportional to the magnetic field and current, and depends inversely on the thickness of the material. The proportionality constant is called the Hall coefficient. The Hall coefficient and resistivity of the material can be related to the material properties carrier density and carrier mobility. The sign of the Hall voltage is the same as the sign of the charge of the carriers and thus provides a determination of carrier type (holes or electrons).
In an ideal geometry, the measured Hall voltage is zero when the applied magnetic field is zero. The voltage measured in a practical experiment, however, is typically not zero, and includes a misalignment voltage and a thermal electric voltage. The misalignment voltage is proportional to the resistivity of the material, the current, and a factor that depends on the geometry. This factor converts resistivity to resistance between two Hall voltage probes. The thermal electric voltage arises from contacts between two different materials and is dependent on the thermal gradients present.
One method for measuring Hall effect involves application of a DC magnetic field to piece of material. In this method, the effects of the thermal electric voltage can be removed by changing the direction of the current applied across the material and measuring the voltage for both current directions. Thermal electric voltage does not depend on current. Thus, the effect of the thermal electric voltage is removed by subtracting the measured voltage at the two different currents.
Similarly, using a DC magnetic field, the misalignment voltage can be removed by changing the direction of the magnetic field and measuring the voltage for both field directions. Misalignment voltage does not depend on magnetic field. Thus, the effect of the misalignment voltage is removed by subtracting the measured voltage at the two different currents.
While the DC-field method may be advantageous for high-mobility materials, it often fails to provide accurate measurements for low-mobility materials, i.e., materials with a mobility less than 1 cm2/Vs. In low-mobility materials, the difference between the voltage measurements at each field direction becomes much smaller compared to the misalignment voltage. As a result, noise in the system can dominate the measurement and produce inaccurate results. Accordingly, Hall measurements using the DC-field method on low-mobility materials often give inconsistent values and carrier signs.
To increase the accuracy of measurements on low-mobility materials, larger magnetic fields are sometimes used, the reversal of which can take a long time, e.g., several minutes, depending on a magnet's configuration. During the time it takes to reverse the DC field, the temperature of the measured material may change. A change in temperature will cause a change in the misalignment voltage between measurements, leading to an incorrect result when the two measurements are subtracted.
Another method of measuring Hall effect uses an AC magnetic field. Because the misalignment voltage is not dependent on the magnetic field, it will remain a DC voltage while the Hall voltage is an AC voltage. A DC current is still used in this method, meaning that the thermal electric voltage also remains a DC voltage. A lock-in amplifier can then be used to easily separate the AC and DC voltages and thus the misalignment voltage from the Hall voltage.
The AC-field method solves some of the problems inherent in the DC-field method. Particularly, the separation of voltages in frequency space allows for much easier detection of smaller voltage differences, for example when measuring low-mobility carriers.
The AC-field method, however, suffers from a large drawback: in actual use it can be slow. The speed is determined by the frequency of the magnetic field. For real magnets and power supplies, the highest operating frequency can be approximately 0.1 Hz (10 seconds per cycle), as determined by the inductance of the magnet. Typically, a lock-in amplifier uses 6 cycles of input for best averaging of the output signal. Thus, a single Hall voltage measurement may take 60 seconds. Further, adding current reversal, which is required to remove the thermal electric voltages and phase errors, a total of 6 measurements are required, making the time to measure a Hall voltage 6 minutes. In practice, this Hall measurement is repeated from 10 to 100 times to get average values and an estimation of statistical variations. In sum, full measurement of a single material can take up to 10 hours under this method.
If the carrier density of a material is large, the Hall voltage will be small. To measure such materials, increasing the current will increase the Hall voltage. However, when the current is increased, self-heating can cause the temperature of the material to change. Specifically, the sample and contacts will have some resistance and power will be dissipated in the sample. This power will heat up the sample. The misalignment voltage is proportional to the resistivity of the material. As the temperature of the sample changes, the resistance changes and hence the misalignment voltage changes. In effect, a term that was a DC signal becomes a time dependent signal that will be detected by the lock-in amplifier in addition to the Hall voltage. The slowly-changing misalignment voltage will look like noise in the measurement. Additional noise comes from the Johnson-Nyquist noise due to material resistance. These two noise contributions to the measurements limit the ability of the AC field Hall method to measure small Hall voltages.
Accordingly, a system and method is needed for more quickly and accurately measuring Hall voltages, particularly for low-mobility materials.